Asymptotic behaviour of bigraded components of local cohomology modules

TL;DR

This paper investigates the long-term behavior of bigraded components of local cohomology modules over polynomial rings with a focus on stability and properties, especially for binomial edge ideals.

Contribution

It provides new insights into the asymptotic behavior and stability of bigraded local cohomology components, particularly when the base ring is regular and for binomial edge ideals.

Findings

Bigraded components exhibit specific asymptotic properties.

Stability of invariants is established under regularity assumptions.

Properties of local cohomology components are characterized for binomial edge ideals.

Abstract

Let $C$ be a commutative Noetherian ring containing a field $K$ of characteristic zero. Let $R=C[X_1, \ldots, X_n, Y_1, \ldots, Y_m]$ be a polynomial ring over $C$ with $\mathrm{bideg}~ c=(0,0)$ for all $c \in C$, $\mathrm{bideg}~ X_i=(1,0)$ and $\mathrm{bideg}~ Y_j=(0,1)$ for $i=1, \ldots, n$ and $j=1, \ldots, m$. Let $I$ be a bihomogeneous ideal in $R$. In this article, we study asymptotic behaviour of bigraded pieces of the local cohomology module $H^i_I(R)$. Moreover, under the extra assumption that $C$ is regular, we investigate the asymptotic stability of invariants associated to its bigraded components. Consequently, we obtain certain properties of components of the bigraded local cohomology module $H^i_I(R)$, where $C=K$ is a field and $I$ is a binomial edge ideal.